t Test for Two Matched Samples

1
Chapter 20

• t- Test for Two Matched Samples

2
Vodka Whiskey
Previous Experiment
Previous Decision
The researcher randomly selected 10 Cranberry
vodka drinkers and 7 Boilermaker drinkers. The
vodka group had a mean drunkenness of 52 and the
whiskey group had a mean drunkenness of 39.
The drunkenness of vodka drinkers and whiskey
drinkers is different
Question Who typically drinks Cranberry Vodkas
and who drinks Boilermakers?
2nd Question Who typically weighs more?
3rd Question Was the difference in the previous
experiment due to different alcohol or different
weights?
3
Two Independent Sample t-Test

• Factors that affect the standard error (s ?1 -?2)
• Population Standard Deviation
• Random Sampling and Sample Size
• Variable estimates of the s
• Variability between the selected samples
• Possibility that the samples differ in the other
  factors that affect the measure.
• e.g. Mostly light group vs mostly heavy group
• Always a problem with random selection

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Two Independent Sample t-Test

• Factors that affect the standard error (s ?1 -?2)
• Population Standard Deviation
• Random Sampling and Sample Size
• Variable estimates of the s
• Variability between the selected samples

5
Matched Samples t-Test

• Reducing a Two-Sample test into a Single Sample
  Comparison
• (Single Sample t-Test)

6
Matched Samples t-Test FAQ

• What does matching mean?
• An observation in one sample is paired with an
  observation in the other sample?

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Matched Samples t-Test FAQ

• How does matching happen?
• Pick a matching variable
• Must be correlated to the dependent measure
• e.g., weight and drunkenness
• Randomly select 2 participants who are equal on
  the matching factor
• Randomly assign 1 participant to each group
• Vodka 110lb, 150lb, 190lb
• Whiskey 110lb, 150lb, 190lb

8
Matched Samples t-Test FAQ

• What should matching do?
• Reduce the between sample variability
• e.g., Affected by weight the same
• Reduce the standard error.
• What is special about the matched samples t-test?
• Mathematically reduces the standard error.
• Brings the critical scores closer to zero
• Makes the test more sensitive.

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Matched Samples t-Test FAQ

• What does the test assume?
• Populations have a normal shape
• . or at least the sampling distributions are
  normal
• Homogeneity of Variance
• Population have the same variability
• Matching variable is correlated to the dependent
  measure

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Matched Samples t-Test FAQ

• What is a difference score? Actual dependent
  measure used by the test.
• It is the difference between the paired scores
• D110lb Vodka110lb - Whiskey110lb
• What is the null hypothesis?
• The groups are equal
• Vodka110lb - Whiskey110lb 0
• mD 0

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Matched Samples t-Test FAQ

• Any other requirements?
• Equal sample sizes!!!!!

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Hypothesis Test 11

• Vodka Whiskey
• Matched Samples t-Test

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Vodka/Whiskey Matched t-Test
Are all types of alcohol the same, even if the
proofs are the same? This question was raised by
a researcher who had observed vodka drinkers and
noticed that they seemed to get drunk faster than
whiskey drinkers. To test whether whiskey is the
same or different than vodka, the researcher
decided to compare people who drank 3 Cranberry
Vodkas to people to drank 3 Boilermakers. Since
weight is known to affect drunkenness, the
researcher has matched the samples to make sure
that weight is evenly distributed between the
group (on next slide).What will the researcher
conclude at a .01 level of significance.
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Vodka/Whiskey Matched t-Test
Step 0) Convert to Difference Scores
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Vodka/Whiskey Matched t-Test
Step 0) Convert to Difference Scores
SD / ndif
1 / 6
.16
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Vodka/Whiskey Matched t-Test
Step 0) Convert to Difference Scores
SD / ndif
1 / 6
.16
6.37
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Vodka/Whiskey Matched t-Test

• Step 1) Rewrite the research question
• Does the mean drunkenness of the vodka
  population equal the mean drunkenness of the
  whiskey population.
• Step 2) Write the statistical hypotheses
• H0 mD 0
• H1 mD ? 0

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Vodka/Whiskey Matched t-Test
Is the mean of the vodka pop. the same as the
whiskey pop?

• Step 3) Form Decision Rule
• Draw Normal Curve
• Shade in a
• Mark Rejection Region(s)
• Determine Critical Scores
• Write conditions for rejection H0

Hypothesis H0 mD0 H1 mD ?0
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Vodka/Whiskey Matched t-Test
Is the mean of the vodka pop. the same as the
whiskey pop?
mD mhyp
0
Hypothesis H0 mD0 H1 mD ?0
2.6
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Vodka/Whiskey Matched t-Test
Is the mean of the vodka pop. the same as the
whiskey pop?
Step 4) Calculate Test Statistic
Hypothesis H0 mD0 H1 mD ?0
.06
Based upon a sampling distribution with
2.6
mD 0
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Vodka/Whiskey Matched t-Test
Is the mean of the vodka pop. the same as the
whiskey pop?
Step 5) Make Decision
Step 6) Interpret Decision

• We have no evidence to suggest that the
  drunkenness of vodka drinkers is different from
  whiskey drinkers.

Hypothesis H0 mD0 H1 mD ?0
Based upon a sampling distribution with
2.6
tobt .06
mD 0
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Why Matching Works!

• Why can we assume the standard error is reduced
  (if the matching variable is correlated with the
  dependent measure)

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Why Matching Works
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When Matching Doesnt Work
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Matched Sample t-Test

• Matched Sample t-Test and the correlation
  assumption.
• The test does not know if the matching variable
  is correlated
• Assumes it is correlated because you selected it
• Drops the standard error estimate
• If the matching variable is not correlated
• Standard error has not actually decreased
• .. but the test lowered it anyway.

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Repeated Measures

• The Ultimate Matching

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Repeated Measures

• Repeated Measures
• Using the same participants in both conditions
• Concern Carry-over Effects
• Spill-Over
• Effects of drugs Linger
• Practice effects
• Second time the participant has done the task
• Counter-balancing
• Some participants given the conditions in reverse
  order

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To match, or not to match

• That is the question

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Matched vs Independent Tests
Which test is more sensitive?
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Matched vs Independent Tests
Which test is more sensitive?
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Matched vs Independent Tests
Which test is more sensitive?
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Two Sample Tests

• Final Notes

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Two Sample Tests

• One-Tailed Tests
• Make the population with the larger hypothesized
  mean population 1
• Always an upper-critical test
• Confidence Intervals
• Confident that the difference between the
  population means is within the interval
• . not about the value of the population means